Model • Demand are filled on a FCFS basis. • Demand are backlogged (if one or more components are missing), and are filled on a FCFS basis. • Inventory of each component is controlled by an independent base-stock policy, – Where si is the base-stock level for component i – For each component i, replenishment leadtimes, Li , are i.i.d., with a cdf of Gi Net inventory at time t, I i ( t ) = s i − X i ( t ), i = 1, … m – , where Xi(t) is the number of outstanding orders of component i at time t. • Immediate availability of all components needed for an arriving demand as the “off-the-shelf” fill rate. – Off-the-shelf fill rate of component i, f i = P[ X i + Qi ≤ si ] – Off-the-shelf fill rate of demand type K, f = P[ X i + Qi ≤ si , ∀i ∈ K ] K K – Average (over all demand types) off-the-shelf fill rate, f = ∑ q K f K K 7/6/2004 Ping Xu 4

Performance Analysis X (t ) = ( X 1 (t ),… , X m (t )) • Derive the joint distribution and steady state limit of vector (See “Suppliers/Arrivals Replenishment Orders” diagram in Lu, Song, and Yao paper) – Each component i, the number of outstanding orders is exactly the number of jobs in service in an M iQi / Gi / ∞ queue with Poisson arrival λi and batch size Qi – The m queues are not independent. Given the number of demand arrivals up to t, the X i (t ) s are independent of one another. – 7/6/2004 Ping Xu 5

Performance Analysis Proposition 1: X (t ) = ( X 1 (t ),… , X m (t )) has a limiting distribution. Derive the generating • function of X. In the special case of unit arrival, Qi ≡ 1 , the generating function of X corresponds to a • multivariate Poisson distribution. For each i, X i is a Poisson variable with parameter λi i = (Σ K∈ℜi λ K ) i • The correlation of the queue is solely induced by the common arrivals. If the proportion of the demand types that require both i and j are very small, the correlation between Xi and Xj is negligible. • Level of correlation is independent of the demand rate. • Reducing the variability of leadtime or batch sizes will result in a higher correlation among the queue lengths of outstanding jobs. 7/6/2004 Ping Xu 6

Operations Research: INFORMS

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